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\title{BEC-BCS Crossover with Feshbach Resonance for Three-Hyperfine-Species Model}
\author{Guojun Zhu}
\department{Physics}
\schools{B.S., University of Science and Technology of China, 2001\\
         M.S, University of Illinois at Urbana-Champaign, 2003}
\phdthesis
\advisor{Anthony Leggett}
\degreeyear{2012}
\committee{Professor Gordon Baym, Chair\\Professor Anthony Leggett, Director of Research\\Professor Brian DeMarco\\Professor Scott Willenbrock}
\maketitle

\frontmatter

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\begin{abstract}
The BEC-BCS crossover problem has been intensively studied both theoretically and experimentally largely thanks to  Feshbach resonances which allow us to tune the effective interaction between alkali atoms.  In a Feshbach resonance, the effective s-wave scattering length grows when one moves toward the resonance point, and eventually diverges at this point.  There is one characteristic energy scale, $\delta_c$, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel.  When the many-body energy scale (e.g. the Fermi energy, $E_{F}$) is larger than $\delta_c$, the closed-channel weight is significant and has to be included in the many-body theory.  Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory.  

The current  thesis addresses the above problem in detail. A set of gap equations and number equations  are derived at the mean-field level.  The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, $a_s$, we find that  the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel.  These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently.  Then the correction due to the inter-channel Pauli exclusion is in the  order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, $E_F/\eta$, which can be analyzed perturbatively over the previous corrections.  

Fermionic and bosonic excitation modes are studied.  Similarly as the mean-field result, the basic structure follows that of the single-channel model, and the correction due to the inter-channel Pauli exclusion can be treated perturbatively with expansion parameter in the order of $E_F/\eta$.  In the bosonic excitation, a new out-of-sync phase mode emerges for the two-component order parameters.   It is nevertheless gapped at the the pair-breaking energy.  
\end{abstract}

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To Jie, Ethan and Chloe
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\chapter*{Acknowledgments}
First I would like to deeply thank my adviser, Professor  Anthony J. Leggett.  This thesis would not have been possible without his guidance and patience.  I would like to thank Professor Monique Combescot from Institut des NanoSciences de Paris for her help, kindness and  invaluable advices.  My thanks also go to  Dr. Shizhong Zhang, Dr. Wei-Cheng Lee, Dr. Parag Ghosh and   Douglas
Packard  for their many discussions and suggestions.  Finally I wish to  thank my wife and my family for their constant support and affection.  

Part of this research presented in  this thesis  is supported  by NSF under grant No. DMR 09-06921. 

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%\item[w.f.] wave function.
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%\item[$\psi$] Wave function.
\item[$\alpha$]  Ratio of the closed-channel correlation $h_{2\vk}$ to the normalized two-body closed-channel bound state wave function, $\phi_{\vk}$, at high momentum.  $h_{2\vk} \xrightarrow{\text{high momentum}}\alpha\phi_{\vk}$. (Page \pageref{eq:pathInt2:hphi})
\item[$\gamma_{i\vk}$] Correction of the fermionic excitation spectrum over $\pm{}E_{\vk}$ and $\epsilon_{\vk}+\eta$. (Page \pageref{eq:pathInt2:xiExpand})
\item[$\delta_{c}$] Energy scale of the  detuning from resonance when the closed-channel takes substantial weight. (Page \pageref{eq:intro:deltaC})
\item[$\Delta_{1,2}$] Order parameters in the open-channel and closed-channel. (Page \pageref{eq:pathInt2:identity})
\item[$\zeta$] Small characteristic dimensionless  parameter for the inter-channel Pauli exclusion, $\zeta=\frac{\Delta_{2}^{2}}{\Delta_{1}\eta}$. (Page \pageref{eq:pathInt2:zetaDef})
\item[$\eta$] Absolute detuning between two channels. Note that $\eta=0$ is not where $a_{s}$ diverges. (Page \pageref{eq:intro:ham})
\item[$\kappa$] Momentum scale of the resonant bound state in the isolated close-channel, namely  $\kappa^{2}/2m=E_{b}$.
\item[$\lambda_{1},\,\lambda_{2}$] Two new parameters in the renormalized gap equation for two channels. They describe the inter-channel Pauli exclusion effects between the two channels. (Eqs. \ref{eq:pathInt2:lambda1} and \ref{eq:pathInt2:lambda2} in Page \pageref{eq:pathInt2:lambda1} and \pageref{eq:pathInt2:lambda2})
\item[$\mu$] Chemical potential.
\item[$\phi_{i}$] Bound-state wave functions of the isolated close-channel; especially, $\phi_{0}$ is the one at resonance. (Page \pageref{eq:pathInt2:phi})
\item[$a_{bg}$] The background s-wave scattering length in the open-channel when it does not coupled to the closed-channel. (Page \pageref{eq:intro:abg})

\item[$a_{c}$] Characteristic size of the bound state at resonance ($\phi_{0}$) if the close-channel is isolated. It is proportional  to the inverse of $\kappa$.
\item[$a_{s}$, $a_{s}^{(o)}$] The effective open-channel s-wave scattering length. Subscript ${}^{(o)}$ will be dropped when there is no ambiguity. Alternatively, it is  the single parameter in  the Bethe-Pierels boundary conditions. (Page \pageref{sec:intro:as}, \pageref{eq:intro:asK})
\item[$a_{0}$] Average inter-particle distance, $a_{0}k_{F}\sim1$.
\item[$E_{b}$, $E_{b}^{(i)}$] Binding energy of the $i^{th}$ two-body (bound) eigenstate in the isolated close-channel.  Superscript ${}^{(i)}$ is dropped when referring to the one in resonance. (Page \pageref{eq:intro:sch2})
\item[$E_{\vk}$] Defined as $E_{\vk}=\sqrt{\epsilon_{\vk}^{2}+\Delta^{2}}$, where $\epsilon_{\vk}=\hbar^{2}k^{2}/2m$ is the kinetic energy. $E_{\vk}$ corresponds the elementary fermionic excitation energy for $\vk$ in the single channel.   In the two-channel case, $E_{\vk}$ is defined in the similar way as, $E_{\vk}=\sqrt{\epsilon_{\vk}^{2}+\Delta_{1}^{2}}$, where $\Delta_1$ is the order parameters of the open-channel. However, it is then only the zeroth order   energy for the two elementary fermionic excitation modes. (Page \pageref{eq:pathInt:G0})



\item[$h_{1\vk}$, $ h_{2\vk}$] (Open and closed)-channel anomalous two-body correlations of the many-body system. (Page \pageref{eq:pathInt2:h2})
\item[$k_{F}$] Fermi momentum. $k_{F}=\hbar(3\pi^{2}n)^{1/3}$ in 3D.

\item[$n_{o(pen)}$] Density of the open-channel atoms.
\item[ $n_{c(lose)}$] Density of the  closed-channel atoms.
\item[ $n_{tot(al)}$] Density of all atoms.
\item[$r_{c}$] Potential extension.  All  potentials are taken as zero outside $r_{c}$. 
\item[$\mathcal{V}_{0}$]  Total volume of the system. 

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